Two-Dimensional Defective Crystals with Non-constant Dislocation Density and Unimodular Solvable Group Structure

In Parry and Zyskin (J. Elast. 127:249-268, 2017) we outlined mathematical methods which seemed to be necessary in order to discuss crystal structures with non-constant dislocation density tensor (ddt). This was part of a programme to investigate the geometry of continuously defective crystals and the symmetries of associated discrete structuresone can think of the programme as an attempt to generalize the use of crystallographic groups as material symmetries in non-linear elasticity theory, for perfect crystals, to deal with the case where defects are present.The methods used rely on the following fact: when the ddt is non-constant, (given technical assumptions), there is a Lie group that acts on the set of material points, and the dimension of the group is strictly greater than that of the ambient space in which the crystal resides. So there is a non-trivial isotropy group associated with the group action. We develop ideas, and recap the requisite mathematical apparatus, in the context of Davini's model of defective crystals, then focus on a particular case where the ddt is such that a solvable three dimensional Lie group acts on a two dimensional crystal state. We construct the corresponding discrete structures too.The paper is an extension of Parry and Zyskin (J. Elast. 127:249-268, 2017), where the analogous group was nilpotent.